13.5.6 NEO-DFT(V)‣ 13.5 Nuclear–Electronic Orbital Method ‣ Chapter 13 Specialized Topics ‣ Q-Chem 6.1 User’s Manual

Within the NEO framework, select nuclei are treated quantum mechanically at the same level as the electrons. This removes the Born-Oppenheimer separation between the quantum nuclei and the electrons and naturally includes nonadiabatic effects between the quantum nuclei and the electrons. At the same time, quantizing the select nuclei gives rise to a potential energy surface with fewer nuclear degrees of freedom, which prevents a direct calculation of the vibrational frequencies of the entire molecule. Consequently, diagonalization of a coordinate Hessian in the NEO framework yields vibrational frequencies and accompanying normal modes of only the classical nuclei, with the quantum nuclei responding instantaneously to the motion of the classical nuclei. ¹⁰⁹⁵ Schneider P. E. et al.
J. Chem. Phys.
(2021), 154, pp. 054108. Link Although the fundamental anharmonic vibrational frequencies of the quantum nuclei can be accurately obtained through NEO-TDDFT, ²⁵⁷ Culpitt T. et al.
J. Chem. Phys.
(2019), 150, pp. 201101. Link the couplings between the vibrations of the classical and quantum nuclei are missing. To obtain the fully coupled molecular vibrations, an effective strategy denoted NEO-DFT(V) was developed. ¹³⁶² Yang Y. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 1167. Link ^, ²⁵⁸ Culpitt T. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 6840. Link The NEO-DFT(V) method has been shown to incorporate key anharmonic effects in full molecular vibrational analyses and to produce accurate molecular vibrational frequencies compared to experiments. ¹³⁶² Yang Y. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 1167. Link ^, ²⁵⁸ Culpitt T. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 6840. Link

The NEO-DFT(V) method involves diagonalization of an extended NEO Hessian composed of partial second derivatives of the coordinates of the classical nuclei ( $\mathbf{r}_{c}$ ) and the expectation values of the quantum nuclei ( $\mathbf{r}_{q}$ ). This extended Hessian matrix is composed of three sub-matrices: $\textbf{H}_{0}=(\partial^{2}E/\partial\mathbf{r}^{2}_{c})|_{\mathbf{r}_{c}=% \mathbf{r}_{q}}$ , $\textbf{H}_{1}=\partial^{2}E/\partial\mathbf{r}_{q}\partial\mathbf{r}_{c}$ , and $\textbf{H}_{2}=\partial^{2}E/\partial\mathbf{r}^{2}_{q}$ , where in each case, all other coordinates of the classical nuclei and expectation values of the quantum nuclei are fixed. The extended Hessian has the following structure:

\mathbf{H}_{\text{DFT(V)}}=\begin{bmatrix}\mathbf{H}_{\text{0}}&\mathbf{H}^{% \top}_{\text{1}}\\ \mathbf{H}_{\text{1}}&\mathbf{H}_{\text{2}}\end{bmatrix}

(13.53)

where


$\displaystyle\mathbf{H}_{\text{2}}$	$\displaystyle=\mathbf{U}^{\dagger}\mathbf{\Omega}\mathbf{M}\mathbf{U}$	(13.54a)
$\displaystyle\mathbf{H}_{\text{1}}$	$\displaystyle=-\mathbf{H}_{\text{2}}\mathbf{R}$	(13.54b)
$\displaystyle\mathbf{H}_{\text{0}}$	$\displaystyle=\mathbf{H}_{\text{NEO}}+\mathbf{R}^{\top}\mathbf{H}_{\text{2}}% \mathbf{R}\;.$	(13.54c)

The quantity $\mathbf{R}=d\mathbf{r}_{q}/d\mathbf{r}_{c}$ and the NEO Hessian matrix is $\mathbf{H}_{\text{NEO}}=d^{2}E/d\mathbf{r}^{2}_{c}$ (without the constraint that the expectation values of the quantum nuclei are fixed). In the expression for the $\mathbf{H}_{\text{2}}$ matrix, $\mathbf{M}$ is the diagonal mass matrix, and $\mathbf{\Omega}$ is the diagonal matrix with elements corresponding to the squares of the NEO-TDDFT fundamental vibrational frequencies. ¹³⁶² Yang Y. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 1167. Link $\mathbf{U}$ is a unitary matrix that transforms $\mathbf{\Omega}$ to the target coordinate system and is approximated with the transition dipole moment vectors afforded by a NEO-TDDFT calculation. ²⁵⁸ Culpitt T. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 6840. Link

Diagonalization of $\mathbf{H}_{\text{DFT(V)}}$ produces the fully coupled molecular vibrational frequencies including anharmonic effects associated with the quantum protons. The NEO-DFT(V) method is available for use with the epc17-2 functional or when no electron-proton correlation functional is used. The NEO-HF(V) method, which involves building the extended NEO-Hessian based on the NEO-HF Hessian and inputs from NEO-TDHF, is also available.

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13.5.6 NEO-DFT(V)